The approach taken to estimate reference state values herein is to utilize methods using state variables and reaction data, where available, to calculate missing values. In absence of state or reaction data, the group contribution method is used, with emphasis on attaining data for the closest structural analogues as a base structure. In doing so, the fewest group values are used to modify the base structure as a means of decreasing the probability of error accumulation. The selection of methods used to estimate reference state data are summarized in Figure 2 using the neutral species as examples.

The Δ G f 0   a q values selected for neutral and ionic organic species are shown in Table 1. The Δ G f 0   a q for organic acids and ions for which there were no data available were calculated from the ionization constants (pKa), critically compiled in [33], and Δ G f 0   a q of the respective ion or acid for which there were reliable data. The only acid-anion pair for which Δ G f 0   a q values were unavailable was succinyl thioester. Therefore, estimation of its Δ G f 0   a q was necessary and attained through the addition of the Δ G f 0   a q for the (>C=O) and (–S-) groups to pentanoic acid-pentanoate Δ G f 0   a q for the succinyl thioester acid-anion pair. The Δ G f 0   a q   ( > C = O ) group value was calculated from the difference in the y-intercepts for regression equations between n-alkanones and (n-1) n-alkanes (both taken from reference [18]). The Δ G f 0   a q   ( − S - ) was calculated similarly, from the difference in y-intercepts from dialkylsulfides [34] and n-alkanes [18], with the same number of carbon atoms. Data for other components of the citric acid cycle (ethyl thiol and acetic acid; see Figure 1) are available in the literature [16,23].

The estimation of high temperature and pressure thermochemical properties for cis-aconitate and isocitrate, for the reactions between citrate and α-ketoglutarate, were not calculated due to deficient reference state data. Since citric acid and its ions are the only feasible tricarboxylated base structures available to use for the estimation, this approach was rejected to avoid circularity. Furthermore, the primary goal of estimating the high pressure and temperature parameters for CAC reactions calculated herein is primarily driven to examine these reactions under the near equilibrium conditions that are present in anoxic systems. Extant microbes that use the CAC reductively to fix carbon generally only contain a partial cycle where the pathway is used to supply precursors for biosynthesis via succinyl-Coenzyme A or α-ketoglutarate [2–4,45].

The S⁰aq values selected are shown in Table 1. If S⁰aq values were not available, third-law entropies were calculated from the aqueous enthalpy and Gibbs free energy values along with the sum of entropies of the elemental constituents using Equation (1): (1) S 0 a q = Δ H f 0 a q − Δ G f 0 a q T − ∑ S elements 0

The S⁰aq for some anions in Table 1 were calculated from the S⁰aq of the acid using Δ_ionS for the ionization reactions from Miller and Smith-Magowan [33]: (2) S 0 a q   ( acid ) = S 0 a q   ( ion − 1 ) + Δ i o n S

The aqueous formation entropies for H-α-ketoglutarte⁻¹ and α-ketoglutarte⁻² were estimated by assuming similar Δ_ionS from succinic acid and its mono- and divalent anions (from Shock [23]). The carbonyl group S⁰aq contribution value (54.3 J mol⁻¹) used for oxaloacetic acid and its respective anions was calculated from the difference in S⁰aq between α-ketoglutaric acid and succinic acid from Shock [23]: (3) S 0 a q   ( α - ketoglutaric   acid )   -   S 0 a q   ( succinic   acid ) = S 0 a q   ( > C = O )

To calculate the S⁰aq needed for acetyl thioester the value of S⁰aq (-S-) was calculated from the difference in aqueous entropies between ethyl sulfide (see Appendix A, Equation 21), and n-butane [18].

The S⁰aq (>C=O) value used for succinyl thioester and its anion was estimated from the difference (68.15 J mol⁻¹ K⁻¹) in y-intercepts between n-alkanones and (n-1) n-alkanes, both from Shock and Helgeson [18]. The S⁰aq (>C=O) value used here was chosen since the value derived from an n-alkanone may be more likely to represent the value of a subterminal carbonyl adjacent to a sulfur atom, as opposed to the α-carbonyl adjacent to a carboxylic acid, as was used above for oxaloacetic acid.

The chosen partial molal volume values are shown in Table 1. The group value V⁰ (>C=O) used for the oxaloacetate series was estimated by the difference in V⁰ between α-ketoglutaric acid [36] and succinic acid [35] such that: (4) V 0   ( α - ketoglutaric   acid )   -   V 0   ( succinic   acid ) = V 0   ( > C = O ) .

This V⁰ (>C=O) was used for pyruvic acid and pyruvate as well since it may rather well represent the volume of an α-carbonyl (V⁰ ~5 cm³ mol⁻¹) adjacent to a carboxylic acid, rather than n-alkanone’s carbonyl with a much larger volume of (~14–15 cm³ mol⁻¹ [30,31]).

The V⁰ for fumarate⁻² was used to estimate the V⁰ of fumaric acid and the H-fumarate⁻¹ anion by assuming the same Δ_ionV relationship for fumaric acid and its ions as for succinic acid from Criss and Wood [35], and H-succinate⁻¹ or succinate⁻² anion, from Shock [23], as demonstrated in Figure 2. The same relationship, using the Δ_ionV (13.13 cm³ mol⁻¹) between lactic acid and lactate from Shock [23] was used to estimate the V⁰ for pyruvate from its acid.

The partial molal volume of the (-S-) group for a thioester was estimated by the addition of the value of V⁰ _(-S-) from a value published by Lepori and Gianni [40] to the V⁰ of 2-butanone taken from Shock and Helgeson [18]. The V⁰ of succinyl thioester and its anion were estimated by the addition of V⁰ _(-S-) [40] and V⁰ _(>C=O) to the V⁰ of pentanoic acid and pentanoate, respectively (both from Shock [23]). The V⁰ _(>C=O) value was estimated to be the difference between the V⁰ of n-alkanones and (n-1) n-alkanes (both taken from Shock and Helgeson [18]).

The standard molal isobaric heat capacities at the reference temperature and pressure are shown in Table 1 and estimation procedures in Figure 2. The C P 0 of pyruvate ion was assumed to have the same difference in Δ i o n C P 0 from its acid, as did the lactic acid-lactate pair [23], of 167.4 J mol⁻¹ K⁻¹. The C P 0 for oxaloacetic acid was estimated by adding the C P 0   ( > C = O ) value (−52.0 J mol⁻¹ K⁻¹) from that reported by Cabani et al. [31] to that of malonic acid [23]. For the estimation of C P 0 values for both H-oxaloacetate⁻¹ and oxaloacetate⁻² ions, it was assumed that the difference from the acid was the same as between succinic acid and it respective anions from Shock [23]. This assumption was also used to estimate C P 0 values of the H-mono- and divalent anions of malate, fumarate, and α-ketoglutarate. The C P 0 of fumaric acid was estimated through subtracting the values for the difference (70.7 J mol⁻¹ K⁻¹) in y-intercepts between n-alkanes from n-alkenes (both values being from Shock and Helgeson [18]) from that of succinic acid. For α-ketoglutaric acid, the C P 0   ( > C = O ) value (−52.0 J mol⁻¹ K⁻¹) reported by Cabani et al. [31] was added to that of succninc acid. The C P 0 for succinyl thioester was estimated from addition of the C P 0   ( > C = O ), and the C P 0   ( - S - ) (−81.2 J mol⁻¹ K⁻¹), both calculated in this work, to the C P 0 of pentanoic acid from Shock [23]. The C P 0   ( - S - ) for succinyl thioester was calculated from the difference in y-intercepts of C₂,C₄,C₆ n-diaklysulfides [34] and C₂,C₄,C₆ n-alkanes [23]. The C P 0   ( > C = O ) value (−135.2 J mol⁻¹ K⁻¹) was calculated to be the difference between 2-pentanone and n-butane, both from Shock and Helgeson [18]. This value for C P 0   ( > C = O ) was chosen for succinyl thioester because a carbonyl from an n-alkanone may be more likely to represent the C P 0   ( > C = O ) in a thioester, as opposed to that from a α-carbonyl adjacent to a carboxylic acid. The resulting value of 216.1 J mol⁻¹ K⁻¹ is in close agreement with the sum of group values provided by Cabani et al. [31], using the (-S-) calculated here, of 216.4 J mol⁻¹ K⁻¹. The C P 0 of the succinyl thioester anion was assumed to have the same difference from its acid as that between pentanoic acid and pentanoate [23]. The C P 0 of acetyl thioester was estimated by adddition C P 0   ( - S - ), as described above, to that of 2-butanone from Shock and Helgeson [18].

Starting with 25°C, 0.1 MPa reference state data, calculating the Δ G f 0 a q P , T at the elevated P and T involves the integration of Equation (5) as: (5) Δ G P , T 0 = Δ G P r , T r 0 − S P r , T r 0 ( T − T r ) +   ∫ T r T C P 0 d T − T ∫ T r T C P 0 d   ln   T + ∫ P r P V 0   d P

The revised-HKF model allows for the incorporation of the changes in the partial molal volume and heat capacity as described in Appendix B.

The partial molal volume of a substance in the revised HKF model is defined by Equation (24). At temperatures of ≤ ~150°C, at the water-saturation vapor pressure (P_sat), the solute-dependent contribution (the non-solvation volume, Δ V n 0) to the partial molal volume term dominates the partial molal volume. The Δ V n 0 term is calculated from Equation (25) utilizing fitting parameters (a₁, a₂, a₃, and a₄) to integrate the Δ V n 0 term into V⁰ at a desired pressure and temperature. The a₁ and a₂ parameters have been generated from empirical data gathered at high pressure and temperature conditions for a variety of compounds [11,15–20,46,47]. The a₁ variable is correlated to a high degree with the Δ V n 0 of a wide range of neutral and charged aqueous organic compounds with a variety of functional groups (Figure 3) and can therefore be used to estimate the values of a₁ for compounds with structural homology to those with high pressure and temperature data for which there are no volumetric data beyond the reference state.

For the range of compounds regressed in Figure 3, the line is defined by: (6) a 1 = 0.5711     Δ V n 0 + 7.4803where a₁ is in J mol⁻¹ K⁻¹ and Δ V n 0 in cm³ mol⁻¹. This regression equation was used to generate the a₁ parameter for all neutral and charged compounds in this work.

The a₂ parameter is somewhat more dependent on the functional group characteristics of a particular molecule. In Figure 4 the Δ V n 0 of aldehydes, hydroxy acids, carboxylic acids, and dicarboxylic acids, and their respective anions, are plotted against a₂.

The data are fit by the line (7) a 2 = 1.341     Δ V n 0   -   16.764where a₂ is also in J mol⁻¹ K⁻¹. For n-alkanones and n-alcohols (Figure 4) the slope is somewhat shallower and the y-intercept lower so these values are fit better by the line (8) a 2 = 1.129     Δ V n 0   -   19.213

The a₂ parameter for all dicarboxylic acids, and dicarboxylate anions, citric acid and its anions, pyruvic acid and pyruvate, and succinyl thioester and its anion were estimated with Equation (7). Since the only apparently large departures from the line in Equation (7) are for shorter- chained carboxylate and hydroxylate anions, a fit for hydroxylates was considered separately to calculate the a₂ parameter for pyruvate. The y-intercept for hydroxylates (Equation (9)) is slightly lower with a steeper slope than Equation (8): (9) a 2 = 1.398     Δ V n 0   -   14.611

However, if the y-intercept value for n-alcohols vs. n-alkanones can be considered comparable to that of the hydroxy and carbonyl groups in hydroxy- and α keto-acids, respectively, then pyruvate’s y-intercept would shift toward the line in Equation (7). Therefore the a₂ value for pyruvic acid using Equation (7) was retained. The a₂ parameter for acetyl thioester was estimated with the line defined by Equation (8) for n-alcohols and n-alkanones.

All a₄ parameters were generated, as suggested by Shock and Helgeson and Shock [18,23], using the correlation with the a₂ fitting parameter. The a₃ parameter was then calculated by solving the rearranged non-solvation molal volume term ( Δ V n 0) of Equation (25) at 25 °C and 0.1 MPa.

The non-solvation and solvation heat capacity contributions to the partial molal heat capacity are defined in Equation (27). The non-solvation contribution term of Equation (27) is expanded in Equation (28) and combines the influence of pressure on the Δ C P , n 0 with incorporation of a substance’s a₃ and a₄ fitting parameters, from above, and the influence of temperature by the use of two heat capacity fitting parameters, c₁ and c₂. The c₂ parameter correlates closely with the reference state C P 0 for compounds with similar functional groups (Figure 5).

Therefore, from this correlation high temperature and pressure data from structurally similar compounds can be used to predict the c₂ parameter of molecules for which no high temperature heat capacity data are available. Figure 5a shows the plot of c₂ vs. C P 0 values, taken from Shock [23], for organic acid anions. The regression of these values for the range of compounds shown is described by the line in Equation (10): (10) c 2 = 0.0507   C P 0   -   17.188where c₂ is in J mol⁻¹. This regression equation was used to estimate the c₂ parameters for all the dicarboxylate and carboxylate anions from the reference state C P 0 used in this study. Equation (11) describes the upper line in the plot of c₂ vs. C P 0 values from organic acids (taken from Shock [23]) in Figure 5b: (11) c 2 = 0.4641   C P 0   -   25.704which was used to estimate the c₂ parameter for all organic acids. The lower line in Figure 5(b) is a plot of the c₂ vs. C P 0 of short-chained n-alcohols and n-alkanones [18] and is described by the line in Equation (12), which was used to estimate the c₂ parameter for acetyl thioester: (12) c 2 = 0.283   C P 0   -   16.970

At temperatures of ≥~150^°C, at the water-saturation vapor pressure (P_sat), the solvent-dependent contribution to the partial molal volume (Equation (26)) and heat capacity (Equation (29)) terms begin to dominate each partial molal function. The conventional (ω) and effective (ω_e) Born coefficients, respectively, are used to describe the substance-specific solvation properties of an ionic species or electrolytes, and neutral species (see Appendix B).

The ω of ionic species has been demonstrated to have a strong correlation with the Δ S f 0 a q [17] and can therefore be calculated from this quantity. In Figure 6 ω is plotted against the Δ S f 0 a q for a variety of mono-, di- and trivalent inorganic and organic anions (taken from Shock and Helgeson (1988) [17]) using relationships in Equations (30–32) to calculate ω, which was also used to calculate the ω for all anions in this work (data also shown in Figure 6).

The effective Born coefficient used to describe the solvation contribution of neutral species in the revised-HKF model has been calculated from S f 0 a q [13,22,23] and the Gibbs free energy of hydration (Δ_hydG⁰) (Appendix C) in the absence of high temperature C P 0 and V⁰ data. It has been demonstrated from empirical data that ω_e generally has a negative value for low- molecular weight neutral organic compounds. The negative value is the result of the inflection of C P 0 and V⁰ values towards positive-infinity near the critical point of water. As the properties of a set of molecules become increasingly polar, the solute-solvent interaction increases. In the case of some neutral inorganic polyhydroxyl compounds such as aqueous silica, of which the hydrated form is thought to be H₄SiO₄, and boric acid (H₃BO₃), both demonstrate electrolyte-like behavior and thus positive ω_e values. The mechanism associated with this phenomenon is thought to arise from water-solute versus water-water competition near the critical point [48], where solutes that are associated with more solvent molecules than is the solvent itself, and will have C P 0 and V⁰ values that approach −∞. The volatility of a substance in comparison to water is also thought to have influence over this process as well. This issue is discussed in detail by Amend and Plyasunov [49], where predictions were made concerning the near-critical point behavior of carbohydrates. It is clear, through their discussion and others, that the relationship between ω_e and solvation at higher temperatures is not an obvious one. For instance, the non-electrolyte amino acid proline displays low-temperature solvation behavior that is best described with a negative value for ω_e. As the temperature increases, however, proline’s solvation is best fit with a positive ω_e. However, this type of solvation behavior may be particular to the proline zwitterion due to its particularly asymmetrical dipole and spatial arrangement of hydrophobic and hydrophilic moieties [22]. If the high density of polar functional groups, as in polyhydroxy compounds, is to be viewed as an indicator of near-critical behavior of compounds with hydroxyl functionalities, then for dicarboxylic acids, oxalic acid with an ω_e that displays neutral behavior may be the closest analogue.

In Table 1, different values for ω_e are displayed as calculated from the correlations with the Δ_hydG⁰ (Equation (44)) and entropy (Equation (45)), as has been done previously. Also included is a systematic estimation of ω_e using values provided by Shock and Shock and Helgeson [18,23] with results shown in Figure 7 in comparison to other series of neutral organic compounds.

The values of these estimations were used for ω_e in all the neutral compounds in this work. The ω_e for pyruvic acid was estimated by taking the ω_e from lactic acid and subtracting the difference between n-propanol and acetone: (13) ω e   lactic   acid   −   ( ω e   n - propanol   −   ω e   acetone ) = ω e   pyruvic   acid

The ω_e for oxaloacetic acid was estimated from the value for malonic acid with addition of the carbonyl group value such that: (14) ω e   malonic   acid   +   ( ω e   2 - pentanone   −   ω e   n - butane ) = ω e   oxaloacetic   acid

Malic acid’s ω_e value was estimated by from succinic acid from: (15) ω e   succinic   acid   +   ( ω e   hydroxybutanoic   acid   −   ω e   butanoic   acid ) = ω e   malic   acidto add the value of the hydroxyl group. For fumaric acid the difference in n-alkanes and n-alkenes were used to modify succinic acid as: (16) ω e   succinic   acid   +   ( ω e   n - butene   −   ω e   n - butane ) = ω e   fumaric   acid

The ω_e for α-ketoglutaric acid also used succinic acid and carbonyl value for oxaloacetic acid: (17) ω e   succinic   acid   +   ( ω e   2 - pentanone   −   ω e   n - butane ) = ω e   α - ketoglutaric   acid

Since there are currently no analogous compounds (tricarboxyl) for citric acid, its ω_e was estimated to be the same as a C₆ compound using the slope and y-intercept for dicarboxylic acids. The value for succinyl thioester was estimated by: (18) ω e   hexanoic   acid   +   ( ω e   2 - pentanone   −   ω e   n - butane )   + ( ω e   diethyl   sulfide   − ω e   n - butane )   =   ω e   succinyl   thioesterusing the value from [22] for diethyl sulfide. The ω_e estimation for acetyl thioester also used the same value for diethyl sulfide: (19) ω e   diethyl   sulfide   +   ( ω e   2 - pentanone   −   ω e   n - butane )   − ( ω e   alkane   − ω e   alkane ( n − 1 ) )   =   ω e   acetyl   thioester .

In consideration of consistency with previous works and the pragmatic aspects of the temperature ranges to be expected to be relevant for these compounds, negative values for ω_e were chosen. In addition, for practical purposes, since the pKa of the strongest acid among these compounds is 2.49 (pyruvic acid from reference [33]), at pH above 3–4 the acids of these compounds will not usually be germane in writing reactions.

Since the motivation for this work is to estimate the high temperature and pressure thermochemical parameters for compounds for which there was no data, the credibility of the resulting values need be examined. The effects of errors on the reference state data, as well as for values calculated beyond the reference state, can be analyzed by the comparison of expected calculations, using the estimated values, with calculations from values in which a sensible error is incorporated. Figure 8a demonstrates the influence of the over- and underestimation of ω_e on the partial molal volume of propanoic acid along P_sat calculated using Equations (25) and (26) with revised HKF parameters from Shock [23]. Although the reference state V P r T r 0 is unaffected by ω_e, the error in the calculated V P T 0 increases with temperature as the partial molal volume is influenced increasingly by the solvation term, although at higher pressures the error is diminished (Figure 8b). However, as we are most concerned with providing data for reaction thermodynamics at elevated pressure and temperature, the effects of inaccurate estimation of the Δ G P , T 0 are of primary concern. As can be seen in Figure 8c, the revised-HKF method is quite insensitive to even a 2-fold under- or over estimation of ω_e, with a maximum of ~0.3% relative error at 350°C. The insensitivity to errors from the estimation methods can be further tested by swapping the HKF parameters of one compound for another, while using the original Δ G f 0 a q. To demonstrate this, for compounds in Figure 8d the original Δ G f 0 a q values were retained while the remaining values (S⁰aq, a₁–a₄, c₁, c₂, and Born coefficients) were mutually swapped from another compound: pyruvic acid and propanoic acid, H-oxaloacetate⁻¹ and H-α-ketoglutarate⁻¹, and oxaloacetate⁻² and α-ketoglutarate⁻² (all values are from Table 2 or Shock [23] for propanoic acid) and used to solve Equation (39). The largest error (~4% relative at 350 °C) is encountered when the equation of state parameters from propanoic acid are replaced with those of pyruvic acid. Considering that there is a relative difference (compared to pyruvic acid) of 15%, 24%, and 12% between the S⁰aq, V⁰, and C P 0 of propanoic acid, respectively, this level of error is quite tolerable, as group additivity estimations generally give relative errors of ~5% for these classes of organic compounds [50]. At 150°C the error is approximately half that at 350°C and is likely to fall below the analytical errors encountered when quantifying the concentrations of these substances.

As an example of the usefulness of these data, values of the logarithm of the equilibrium constant (log K) for acid dissociation reactions were calculated at different temperatures from the Δ G P , T 0 values at P_sat using Equations (38) and (39) and: (20) log   K = − Δ G ∘ 2.303 R Twith the partial molal properties and equation of state parameters in Table 1 (Figure 9). These values allow us to evaluate the potential for each of these reactions to occur under different geochemical conditions. The plots in Figure 8 allow investigation of the pH dependence of the speciation among CAC components. Similarly, the data and parameters, along with the revised HKF equation of state, allow evaluation at wide ranges of temperatures and pressures of reaction energetics among species in the various steps of the CAC (see Figure 1) to determine the thermodynamic viability of these reactions for a variety of conditions. If life did indeed begin under hydrothermal geochemical conditions, these calculations can help identify the conditions necessary for this development.